The Buddha offered his own achievement as evidence, combined with the logical argument that suffering has a cause and therefore a cessation.
The Buddha's primary evidence for Nirvana's achievability was his own attainment. In the earliest Buddhist texts, particularly the Pali Canon, he claims to have awakened to Nirvana himself and describes the experience as the cessation of craving and aversion leading to profound peace. He presented himself not as a god or supernatural being, but as a human who had followed a method and reached the goal. This meant Nirvana was not confined to mythological realms or divine beings—it was accessible to ordinary people willing to undertake the discipline.
The Buddha also pointed to his disciples as additional evidence. The texts record that many of his followers achieved Nirvana during his lifetime, including both monastics and lay practitioners. These were not select individuals chosen by fate, but people from various backgrounds and social positions who followed his teachings. Their success demonstrated that the path worked for more than one person under specific circumstances.
Rather than relying solely on testimony, the Buddha built his argument on a logical foundation: the principle of dependent origination. He taught that suffering arises from specific causes—chiefly craving and ignorance. If suffering has causes, then removing those causes must logically result in the cessation of suffering. This reasoning appears in the Four Noble Truths, the Buddha's foundational teaching.
The Buddha invited his followers to test this logic themselves rather than accept it on faith. He explicitly stated in various suttas that his disciples should verify his teachings through direct experience, not blind belief. This empirical approach suggested that the claims about Nirvana could be validated by anyone patient enough to follow the practice, making it theoretically achievable for all.
You correctly identify that the Buddha refused to describe Nirvana positively. In the Pali Canon, he uses primarily negative language: it is the end of suffering, the absence of greed, hatred, and delusion. He rarely describes what Nirvana is like directly, instead describing what it is not. This apparent limitation was actually central to his argument for its achievability.
The Buddha's reasoning was pragmatic: attempting to describe an experience beyond ordinary mental states would mislead people into chasing a conceptual fantasy rather than pursuing actual practice. He compared this to trying to describe salt to someone who has never tasted it—words fail to convey the experience. The point was not that Nirvana was unknowable in principle, but that it could only be known through direct experience, not through intellectual understanding. This made the lack of description evidence of integrity rather than evidence against achievability.
Later Buddhist schools developed varying explanations of Nirvana's achievability, though they generally accepted the basic premise. Theravada Buddhism, closest to the earliest texts, maintained that Nirvana is the unconditioned reality that exists independent of whether anyone experiences it. Mahayana Buddhism introduced concepts like Buddha-nature, suggesting all sentient beings possess the capacity for enlightenment. Tibetan Buddhism preserved detailed meditative practices aimed at realizing emptiness, which it equated with Nirvana.
Despite these differences, all major traditions agreed on one point: the Buddha's own attainment served as the proof that Nirvana is achievable. Whether interpreted as an escape from conditioned reality, the realization of emptiness, or the activation of Buddha-nature, the fact that someone had reached it established it as a genuine possibility.
The Buddha's ultimate answer to your question shifts the burden from theoretical proof to practical method. He essentially argued: 'I achieved it. My followers have achieved it. Here is the step-by-step path. Follow it yourself and judge whether the goal is achievable.' This places Nirvana's evidence not in words or concepts, but in the verifiable success of the practice.
This approach acknowledges a fundamental truth the Buddha taught: that direct experience trumps all intellectual theories. The impossibility of describing Nirvana in advance does not undermine its achievability; rather, it underscores that achievement requires doing the work yourself rather than understanding it mentally first.